By Proposition REF -(ii) and Proposition REF -(ii), using the notion of refined Gysin homomorphism which is generalized to the case of Deligne-Mumford stacks in [[1]}] and to the case of Artin stacks in [[2]}], as in [[3]}, Section 6.2], one has the localized first Chern class \(c_{1}^{\mathrm {loc}}(L_{\varrho },s_{\varrho })\)  satisfying
\([\mathfrak {PT}_{\mathfrak {X}_{0}^\dagger /\mathfrak {C}_{0}^\dagger }^\varrho ]^\mathrm {vir}=c_{1}^{\mathrm {loc}}(L_{\varrho },s_{\varrho })[\mathfrak {PT}^P_{\mathfrak {X}/\mathfrak {C}}]^\mathrm {vir}\) 
